will it be necessary to take the extra trip to Sydney to renew her US passport?
Apparently you could also go to Melbourne or Perth. The embassy says that "port of entry officials in the United States do have the authority to issue fines should you choose to disregard this law," but I'm not sure that is true. Even if it is true, they apparently do not actually issue them, although that could change without notice. See What is the penalty for US citizens entering/leaving the US on a foreign passport?
Another thought that may not have occurred to you: if you're planning to travel to another country before going to the US, you could get her passport there instead, or you could arrange a detour to Canada or Mexico on your way to the US for that purpose. I realize this isn't likely to be helpful for most people, but it might be for some.
If I were you I would certainly consider applying for ESTA while disclosing the US citizenship in the application. If the ESTA is granted, though I suppose the chance of that happening is low, try using it. There are some accounts here of US citizens receiving ESTA authorization for their foreign passports, but these may have occurred before the application asked about other nationalities, and in one case the airline did not allow the traveler to board the flight, so use this approach with caution. It's not clear from the latter answer, but it's likely that the problems arose at the US preclearance immigration inspection rather than with the airline's ground agents.
If she does fly with her Australian passport and ESTA, you should probably disclose her US citizenship at the passport inspection desk, and present her expired US passport as evidence. If it's feasible, you could also fly to Canada or Mexico and do this at a land border, so ESTA would not be required.
Once you're in the US, you will have many more options for places to apply for her passport, including any post office.
See also Apply For or Renew My Passport > Children Under 16.